Capítulo 4. El mundo del envase
4.7 Rol del envase y del diseñador
The value of the firm’s operations, denoted by V (P, S, W, δ, t), is a function of the state variables and time, t. It is assumed that at specific fixed times, the firm makes a decision about whether to move to another stage of operation. These discrete decision times are given as follows
Td≡ {t0 = 0 < t1 < ... < tm <, ..., tM = T − ∆t} (2.10)
where we assume that the decision to move to another stage of operation occurs instan- taneously at t ∈ Td. Excluding time T from the decision dates implies that bankruptcy
cannot happen in the final instant. If bankruptcy has not occurred during the project life and the firm reaches time T , it has to clean-up the site.
Choices regarding optimal rates of abatement, a, and extraction, q, are made in con- tinuous time at time intervals given as follows
Tc≡ {(t0, t1), ..., (tm−1, tm), ..., (tM −1, tM)}. (2.11)
Since we search for the closed loop control, we assume the controls are in feedback form, i.e., functions of the state variables. Control variables can be specified as: q(P, S, W, δ, t), a(P, S, W, δ, t); t ∈ Tc, and δ+(P, S, W, δ, t); t ∈ Td. We specify a control set which contains
the controls for all t0 ≤ t ≤ tM as follows
K = {(δ+)t∈Td ; (q, a)t∈Tc}. (2.12)
Regardless of the controls chosen, the value function can be written as the risk neutral expected discounted value of the integral of cash flows, given the state variables, with the expectation taken over the controls:
V (p, s, w, ¯δ, t) = EK " Z t0=T t0=t e−r(t0−t) π(P (t0), S(t0), W (t0), δ) dt0 + e−r(T −t)V (P (T ), S(T ), W (T ), δ(T ), T ) P (t) = p, S(t) = s, W (t) = w, δ(t) = ¯δ # (2.13) where (p, s, w, ¯δ) denote realizations of the random and path dependent variables (P, S, W, δ). r is the risk free interest rate, and E[·] is the expectation operator. The value in the final time period, T , is assumed to be the net benefits from restoring and closing the mine. This is described as a boundary condition in AppendixB.1.
2.3
Dynamic Programming Solution
Equation (2.13) is solved backwards in time using dynamic programming. For a particular tm ∈ Td, we define t−m and t+m to represent the moments just before and after tm. At tm,
we determine the discrete optimal control δ+, while in the interval (t+m, t−m+1). We solve for the optimal controls q and a in continuous time.
2.3.1
Determining optimal rates of abatement, a, and extraction,
q, from t
−m+1→ t
+m
Using a standard contingent claims approach (Dixit and Pindyck, 1994), we can derive a system of partial differential equations that describe the value of the resource, V , in the interval (t+
m, t −
m+1) for all operating states except for abandonment.
∂V ∂t + 1 2σ 2p2 ∂ 2V ∂p2 + κ(ˆµ − ln p)p ∂V ∂p + max q,a n − q ∂V ∂s + (φq − a) ∂V ∂w + π(t) o − λ(·) Vbankrupt− V + rV = 0, for δ = δi, i = 1, 2, 3 (2.14)
where we maximize with respect to the control variables a and q. The hazard function, λ(·), is given by Equations (2.5) and (2.6). We set the value of the project after bankruptcy to zero – i.e., Vbankrupt = 0. Therefore, V represents the project value prior to bankruptcy.
Let LV be the differential operator as follows LV = 1 2σ 2p2 ∂2V ∂p2 + κ(ˆµ − ln p)p ∂V ∂p + r + λ(·)V. (2.15) Substituting LV in Equation (2.14) gives
∂V ∂t + LV + maxq,a n − q ∂V ∂s + (φq − a) ∂V ∂w + π(t) o = 0, for δ = δi, i = 1, 2, 3 (2.16)
Once the project is in stage 4, the project value goes to zero.
V (p, s, w, δ = δ4, t) = 0. (2.17)
Note that we analyze the firm’s investment decisions under the Q measure which allows us to use a risk-free interest rate as well as risk-adjusted parameters for the commodity price and for the probability of bankruptcy. The risk due to price volatility can be hedged by deducting the market price of price risk from the drift rate (see Section (2.5)). This market price reflects an additional return over the risk-free interest rate that the firm demands per unit of price volatility.
As is discussed in Insley and Lei (2007) and in Ayache et al. (2003), there are two approaches to hedging a jump risk due to bankruptcy (or other causes). One is to assume the risk is fully diversifiable in a portfolio of assets. In this case the asset would generate no extra return for an investor due to bankruptcy risk and it can be assumed that the real world probability of bankruptcy is equal to the risk-neutral probability. The second approach is to assume that the risk of bankruptcy can be hedged by trading another asset which faces the same risk. In this case, the market price of a jump-related risk (i.e., bankruptcy risk in our study) will be used in the valuation model instead of the actual probability (i.e., the historical probability of bankruptcy in our study). This implies that, in our study, λ(·) in Equation (2.14) should be replaced by the market price of bankruptcy risk reflecting an additional return over the risk-free interest rate that the firm requires to obtain per each unit of potential loss in the project value due to bankruptcy.
The market price of bankruptcy risk reflects the market’s perception of the bankruptcy risk and could be higher from the historical bankruptcy risk. It has been observed that the corporate bond yields exceed the risk-free rate by an amount greater than what is justified by historical default rates (Amato and Remolona, 2003).
This study assumes that the risk from price volatility can be hedged. In Chapter 1, the risk-adjusted parameters of the commodity price including the market price for the price risk are estimated using futures prices. In Section (2.5), we have used the same estimated values for the price parameters in this study. However, estimating the market price of bankruptcy risk is beyond the scope of this paper and instead we examine the sensitivity
of our results to the parameters of the hazard function in Appendix B.2.
2.3.2
Determining optimal operating stage, δ at t
mFor tm ∈ Td, the firm checks to determine whether it is optimal to switch to a different
operating stage. The firm will choose the operating stage which yields the highest value net of any costs of switching. Let C(δ−, δ0) denote the cost of switching from stage δ− to δ0. Recall that t = t− represents the moment before tm and t = t+ denote the instant after
tm. Solving going backward in time, and noting the optimal stage is denoted as δ+, the
value at t−m is given by
V (p, s, δ−, t−m) = V (p, s, δ+, t+m) − C(δ−, δ+) (2.18) δ+ = arg max
δ0
[V (p, s, δ0, t+m) − C(δ−, δ0)].
Switching costs are the same under the bond or strict liability policies except for project commencement as well as when the mine is abandoned. Opening the mine under the bond requires investment cost and initial bond payment, whereas the latter is absent under the liability rule. In addition, recalling from Assumption 10, the abandonment cost with bonding requirements, C(δi, δ4), i = 2, 3, simply equals the negative of reimbursement
after clean-up has been completed. However, Assumption 10 indicates that under the strict liability rule, C(δi, δ4), i = 2, 3, equals the firm’s expected clean-up costs. Note that
no waste is created in Stage 1.
2.4
Numerical solution approach
Equations (2.16) and (2.18) represent a stochastic optimal control problem which must be solved using numerical methods. The computational domain of Equation (2.16) is (p, s, w, ¯δ, t) ∈ Γ where Γ ≡ [pmin, pmax] × [0, s0] × [0, ¯w] × {δ1, δ2, δ3, δ4} × [0, T ]. More
details are given in Appendix B.1 where boundary conditions are specified for the PDEs. LV in Equation (2.16) can be discretized using a standard finite difference approach. The
κ 0.0264 (0.001) Root Mean Square Error 0.07 µ 2.7051 (0.079) Mean Absolute Error 0.05
η 2.7845 (0.026) Log-likelihood function 9652 σ2 0.0458 (0.002) Number of observation 937
Table 2.1: Estimation results for the one-factor copper price model using Kalman Filter. RMSE, MAE, µ, and η are in terms of US $/lb. Standard errors are in parenthesis. Weekly futures data from Aug 1st, 1997 to Jul 13th, 2015.
other terms in the equation are discretized using a semi-Lagrangian scheme as described in Chen and Forsyth (2007) and will not be described further here. Recall that the optimal control for q which we denote by q∗ is bang - bang so that q∗ ∈ {0, ¯q}. To jointly determine the optimal controls, (q∗, a∗), we discretize the control a ∈ [0, ¯a] and determine the optimal controls by exhaustive search at each point in the state space (p, s, w, t).
2.5
An application to the copper industry
To compare the impacts of bankruptcy on the firm’s optimal decisions under each policy with the results of Chapter 1 obtained for a solvent firm, this study considers the case of investment decisions for a copper mine similar to Chapter 1. The parameters of the stochastic model assumed for copper prices are already estimated in Chapter 1 using weekly data for copper futures contracts that are traded in London Metal Exchange Market from August 1997 to July 2015. These estimations are summarized in Table (2.1) and are obtained by adopting a Discrete Kalman Filtering approach and a Maximum Likelihood Function as explained in Schwartz (1997). Note that we define the parameter ˆµ = µ − η so that the market price of price risk, η, is deducted from µ which is the long-run mean of ln(P ) before adjusting for the price risk. The market price of price risk, η, reflects additional returns that the firm demands over the risk-free interest rate per each unity of price volatility. A numerical example is developed based on available data from an open-pit copper mine in British Columbia, supplemented by our assumptions when data is lacking, as given by Table (2.2).
Life of project T = 15 years
Risk-free rate r = 0.02 per year
Initial reserve s0 = 1173 million lb
Strip ratio (waste:ore) φ = 1.5 : 1
Production capacity q = 78.2¯ million lb/year
Abatement ceiling a = 2φ¯¯ q million lb/year
Landfill capacity w = 2200¯ million lb
Extraction cost Cq(q) = γq γ = 1.35 $/lb
Abatement cost Ca(a) = αa2 α = 10−3
Firm’s clean-up cost Cf(w) = βw2 β = 10−5
Adjustment factor ν = 1.30
Hazard function (Scenario I) λ(p, w) = k0
p k0 = 10
−1 Hazard function (Scenario II) λ(p, w) = k1+k2w
p k1 = 10
−1
k2 = 1.5 × 10−4
Project stages δ1, δ2, δ3, δ4
Fixed decision time τd every year
Construction cost I $385 million
Cost to mothball and reactivate C(δ2, δ3), C(δ3, δ2) $5 million Fixed costs of sustaining capital C2m, Cm1
3 $1.66 million/year
Fixed monitoring costs while mothballed Cm2
3 $1 million/year
Table 2.2: Parameter values and functional forms for the prototype open-pit copper mine. All dollar values are based on 2007 US dollars.
2.6
Results analysis
This section compares the impacts of the environmental bond and the strict liability rule on the firm’s optimal investment decisions indicated by critical prices, under each scenario. In addition, we compare the project value and optimal abatement decisions, under each policy and each scenario, at the initial time. Results are presented for Scenario I (bankruptcy risk depends only on commodity price) and Scenario II (bankruptcy risk depends on commodity prices and the level of waste), as well as for no bankruptcy risk. Recall that in the case examined, it is assumed the firm receives the risk-free rate from the government on the bond and also that the firm finances the bond from its retained earnings. This was referred to as Case I bond in Chapter 1. All figures in this section are shown for time zero and
reserves at maximum level.
2.6.1
Valuation results
Figure (2.2) illustrates the value of the investment project prior to construction across different levels of starting prices and initial waste, when the reserve is fixed at its initial level. The top panels are for the strict liability rule and the bottom panels represent the bonding policy. Scenarios I and II are shown in the left-hand panels and the right- hand panels, respectively. Whether for the bonding policy or the strict liability rule, an increase in the initial waste generated through construction reduces the value of the project. As noted in Chapter 1, higher initial waste reduces the remaining capacity of the landfill. In addition, the waste will have to be cleaned up upon project termination. With no bankruptcy risk, this expected clean-up cost is the same under the either policy. Including the risk of bankruptcy reduces the project value compared to the no bankruptcy case over all values of the waste pile. Such lower values are to be expected since the exogenous bankruptcy risk may cause early termination of the project under the bond, and may increase the during of inactivity under the liability. This behaviour is discussed in Section (2.6.3).
Figure (2.3) compares the value of project with no bankruptcy versus Scenarios I and II for a particular price level. In this figure, we observe that the two policies no longer give the same result. Project values under both Scenarios I and II under the bond are less than for the liability policy. This result follows because under the bond the firm must pay the clean-up costs up-front. If bankruptcy occurs, the firm will not receive a refund on the bond. In contrast, under the strict liability rule, bankruptcy would allow the firm to avoid paying the clean-up costs. Under the possibility of bankruptcy, the bond is much more costly to the firm. This figure also highlights the difference in the firm value under the two scenarios. When the risk of bankruptcy depends on the waste stock under Scenario II, we observe lower values compared to Scenario I. We also observe, not surprisingly, that the value of the project decreases much more markedly as the waste stock builds up in Scenario II.
0 1000 2000 2000 10 US$ million 3000
Solution Surface at t = 0, Liability (Scenario I)
w 1000 p 4000 5 0 0 0 1000 2000 2000 10 US$ million 3000
Solution Surface at t = 0, Liability (Scenario II)
w 1000 p 4000 5 0 0 Vb(p0=10,w0=0)=3345 Vb(p0=10,w0=2200)=2540 Vb(p0=2, w0=2200) =11.64 Vb(p0=10,w0=0)=3427 Vb(p0=10,w0=2200)=3286 Vb(p0=2, w0=2200) =46.62 Vb(p0=2,w0=0)=54.71 Vb(p0=2,w0=0)=61.42 0 1000 2000 2000 10 US$ million 3000
Solution Surface at t = 0, Bond (Scenario I)
w 1000 p 4000 5 0 0 0 1000 2000 2000 10 US$ million 3000
Solution Surface at t = 0, Bond (Scenario II)
w 1000 p 4000 5 0 0 Vb(p0=2,w0=0)=61 Vb(p 0=2,w0=0)=54.46 Vb(p0=2, w0=2200) =43.18 Vb(p0=2, w0=2200) =9.682 Vb(p0=10,w0=2200)=3265 Vb(p0=10,w0=2200)=2498 Vb(p0=10,w0=0)=3344 Vb(p0=10,w0=0)=3424
Figure 2.2: Project value prior to construction, given s0 = 1173 million pounds, for all price levels and waste piles, under the liability rule (the top panels) and the bonding policy (the bottom panels), for Scenario I (the left-hand panels) and Scenario II (the right-hand panels). w: million pounds and p: US$/pound.
0 500 1000 1500 2000 2500 0 20 40 60 80 100 120
Comparing project values prior to construction at p
0=2 Waste Pile US$ millions Liability, Scenario II Liability, Scenario I Bond, Scenario I Bond, Scenario II Bond & liability, solvent firm
Figure 2.3: Project values prior to construction across waste pile (million pounds), under Sce- narios I and II versus no bankruptcy case, at p0 =US$2/pound and s0= 1173 million pounds.